We study the volatility time series of 1137 most traded stocks in the US stock markets for the two-year period 2001-02 and analyze their return intervals $tau$, which are time intervals between volatilities above a given threshold $q$. We explore the probability density function of $tau$, $P_q(tau)$, assuming a stretched exponential function, $P_q(tau) sim e^{-tau^gamma}$. We find that the exponent $gamma$ depends on the threshold in the range between $q=1$ and 6 standard deviations of the volatility. This finding supports the multiscaling nature of the return interval distribution. To better understand the multiscaling origin, we study how $gamma$ depends on four essential factors, capitalization, risk, number of trades and return. We show that $gamma$ depends on the capitalization, risk and return but almost does not depend on the number of trades. This suggests that $gamma$ relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of $tau$, $mu_m equiv <(tau/<tau>)^m>^{1/m}$, in the range of $10 < <tau> le 100$ by a power-law, $mu_m sim <tau>^delta$. The exponent $delta$ is found also to depend on the capitalization, risk and return but not on the number of trades, and its tendency is opposite to that of $gamma$. Moreover, we show that $delta$ decreases with $gamma$ approximately by a linear relation. The return intervals demonstrate the temporal structure of volatilities and our findings suggest that their multiscaling features may be helpful for portfolio optimization.
Download